Sounds like he is bluffing. You need to find a support group of others in your circumstances.
They will have access to any legal advice you need, and what his ACTUAL rights are.
Court costs aren't cheap so unless he has unlimited funds he can't be up too much.
The thing with the dog is just a NO. He doesn't have the right to enforce that on you either.
I don't know enough about the laws where you are.
And it doesn't sound like you are sure either.
You must find out. Don't listen to him or try and guess based on others. All situations are different.

He will try and frighten and manipulate you by claiming to know the law, and that you don't have a chance.

Please please, go and look into it. There are a lot of single mothers support groups. And I am sure if you ask around people can point you in the right direction of finding out your rights.

Don't just wait and see, keeping your fingers crossed, because it's much harder to reverse a court order than prevent one in the first place.

Good luck.